The successful detections of gravitational waves (GWs) bring some new light to cosmological research. Through the analysis for the GW's waveform, absolute luminosity distance can be measured, which is extremely important for cosmology and referred to as standard siren. The campaign in observing the event of binary neutron star (BNS) merger (known as GW170817 [1]} as a GW event) by detecting GWs and electromagnetic waves (EMWs) in various bands opened a new era of multi-messenger astronomy [2]}. The observations of GWs and their EM counterparts enable us to establish an absolute luminosity distance–redshift relation, which can play an extremely important role in breaking the cosmological parameter degeneracies generated by the traditional EMW cosmological probes [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}. Nevertheless, even though the GW standard sirens are used, the information loss still exists in inferring the EoS of dark energy due to the two integrals in the expression of distance relating to \(w(z)\) . Thus, the best way of constraining \(w(z)\)  is to directly measure \(H(z)\) , instead of the distance–redshift relation. Actually, the ways of directly measuring \(H(z)\)  by using GW observations have been proposed.
